Weird connecting lines with no edges/vertices
This is because you have a non-planar ngon, i.e. a non-flat face with more than four sides. This is particularly problematic with an ngon as blender has no indication of what the topology should be inside the face, resulting in ambiguous geometry. For display purposes there must be something there, so it takes a guess.

You should try to avoid such faces, as they will not behave well with subsurf or deformations (to name a few issues).

To fix this, you can either 


Delete the face and fill in the hole with planar quads. Grid fill might be of help here.
Triangulate the face (&#9096; CtrlT), splitting the large ngon into multiple planar triangles:



You could also try Mesh > Cleanup > Split non-planar faces.

Note that this still won't subdivide or deform nicely.